Abstract
Let M be a most singular orbit of the isotropy representation of a simple symmetric space. Let \((\nu _i, \Phi _i)\) be an irreducible factor of the normal holonomy representation \((\nu _pM, \Phi (p))\). We prove that there exists a basis of a section \(\Sigma _i\subset \nu _i\) of \(\Phi _i\) such that the corresponding shape operators have rational eigenvalues (this is not in general true for other isotropy orbits). Conversely, this property, if referred to some non-transitive irreducible normal holonomy factor, characterizes the isotropy orbits. We also prove that the definition of a submanifold with constant principal curvatures can be given by using only the traceless shape operator, instead of the shape operator, restricted to a non-transitive (non necessarily irreducible) normal holonomy factor. This article generalizes previous results of the authors that characterized Veronese submanifolds in terms of normal holonomy.
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Supported by Famaf-UNC, CIEM-Conicet, Argentina/ Colciencias and Universidad de Los Andes, Colombia.
This research started during the visit of the first author to the Universidad de Los Andes and was essentially finished during the visit of the second author to FaMAF.
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Olmos, C., Riaño-Riaño, R. Normal holonomy and rational properties of the shape operator. manuscripta math. 157, 467–482 (2018). https://doi.org/10.1007/s00229-017-0993-9
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DOI: https://doi.org/10.1007/s00229-017-0993-9